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Kz=J ozW, TO no BTopoii H3 cpopMyJI (90)

M~z= - J0 z·W= -loz•8,

(91)

r)J.e e - yrJiosoe ycKopeHHe TeJia.

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C TeJia, H nape c MOMeHTOM M~z=-Jcz ·e.

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§ 138. 803MO)I(HbiE flEPEMEU(EHH~ CHCTEMbl. 'IHCJIO CTEflEHEft CBOISO~bl

3$lJeKT MeXaHHqecKHX CB513efi MO>KHO. ytiitTb!BaTb, He TOJibKO BBO~ll

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H3MeHHTbCli BHlJ.. pacCM3TpiiBaeMOH MeXaHI-f'IeCKOH CHCT€Mbl.

HanpnMep, B KpnoomiiiiHo-noJI3YHHOM MexaHR3Me, nao6paJKeHHOM HIIJKe na pHc. 356 (eM. § 140), nepeMemenne na noKa3aHnoro noJiolKeHnH B noJioJKerme, npa KOTOpOM cp=(}, HeJib3H p3CCM3T!J'IiBa'!'bK3K B03MOl*HOe, T3K KaK npn cp=Q ~KT HaJIOJKeHHb!X CBH3eii 6y.n:eT .llPYrY!M, qTO, B qaCTHOCTii, li3MeHiiT yc.10Biie paBHOBeCHH MexaHH3Ma no.u. ,!l;eikTBHeM cnJlbl. P u napb! c MoMeHTOM M. Toqno TaK lKe HeJih3H cquTaTb B03MOlKiib!M }.l.alKe 3JleMeHT3j;)HOe nepeMeiii,eHiie TOqKR B maTyHa B,llO.llb JIH• HUH AB; ono 6b!JJO 6bi B03MOJKHb!M, ecJIU B TOqKe B BMecTo noJI3yHa 6biJia 6bl Kaqa!d· maHcH MyqlTa (pnc. 161 B §57, MycpTa C}, T. e. xor.n:a MexanusM 6b!JI 6bl ll.PYrHM.

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IIOHHM3'fbTe B03MQ)KHble nepeMern,eHHH, npii KOTOpbiX CBH3H COXpaHJI• IOTCR (TOtJKH CHCTeMbl OT CBSl3eH He «OCB06om~a!OTCH»).

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123): uaeaAbflbl.MU Ha3bl8Q/OmcH.

oeAflHJtt{UX tWJJO.JICeHue cucme.Mbl, coBnaaaem c <tucAO.M ee cmeneHeu CBo·

6oabt. fiOolTOMY y T3KOH CHCTeMhl lJHCJIO CTeneHeH CB060,ll.bl MOJKHO onpe- ,!l.eJIHTb KaK IIO lJHCJIY He3aBHCHMbiX B03MOJKHbiX nepeMeru.eHHH, TaK It no lJHCJIY HesaBHCHMhiX Koop,!l.HHaT. TaK, y KPHBOIIIHnHo·nOJI3YHHoro MeX3HI13M3 (eM. HHJKe puc. 356) O,!l.H3 CTeiTeHb CB060,!l.bl (y Hero O,ll.HO He33BHCHMOe B03MOJKHOe nepeMeru.eHHe, HanpHMep IIOBOpOT KpHBOIIIHlla

OA, H O,ll.Ha HesaBHCHMaH Koop,!l.HHara, HanpnMep yroJI fP). Y cBo6o)l,- HOro TBep,ll.OrO TeJia IIIeCTb CTeiieHefi CB060)l.bl (He3aBHCHMbiX nepeMeru.e· Hnii - TpH nocTyrraTeJibHhiX B,!l.OJih Koop.u.mJaTHbiX oceii H TPH noBopoTa BoKpyr 3THX oceU:, a HesaBHCHMbiX Koop,ll.HHaTTPH Koop,!l.HHaTbl nomoca H TPH yrJia 3iiJiepa).

§ 139. UPHHU.HU B03MO)I(Hb1X nEPEMEU.tEHHA

flepeii,!l.e.M K paCC.MOTpeHHIO eru.e O)l.HOro npHHIJ,Hfia MexaHHKH, KOTOpblll ycTaH3BJIHBaeT o6w,ee yCJIOBHe paBHOBeCMH MeXaHHlJeCKOH CllCTeMbl. flo,ll. paBHOBeClfeM (eM.§ 1) Mbl IIOHHMaeM TO COCTOHHHe CHCTeMbl, TipH KOTOpoM BCe ee TOlJKI:l IIO,ll. ,ll.eHCTBHeM npHJIOJKeHHb!X CHJI H3XO)l.5ITCH B IIOKOe no OTHOIIIeHHIO K HHepii,HaJibHOH CHCTeMe OTClJeTa (paccMaTpHBaeM TaK Ha3hiBaeMoe «a6co:rnoTHoe» paBHoBecue). O_uHoBpeMeHHO 6y- ,!l.eM ClJHTaTb BCe HaJIOJKeHHbie Ha CHCTeMy CB.H3H CTal'HJHapHbiMH II crreurraJihHO 3To B ,!l.aJihHei'riiieMKaJK.U.hiH pas oroBapuBaTh He 6y,!l.eM.

BBe,!l.eM noii.HTHe o B03MOJKHoli pa6ore, KaK o6 3JieMeHTapHoii pa6oTe, KOTopyiO ,ll.eHCTBYIOili.3H Ha MaTepHaJibH}'IOTOtiKY CHJia MOrJia 6bi COBepIIIHTh Ha nepeMeru.eHHH, COBTia)l.aiOill.eM C B03MOJKHbiM nepe:\1elll.eHHeM

~noli TOlJKH. Ey,!l.eM B03MOJKHYIO pa6oTy aKTHBHoii CHJibi f• o6osHalJaTb CHMBOJIOM tJA a (6A •=p• ·6r), a B03MOJKHY!O pa6oTy peaKIJ,lUI N CBH3HCHMBOJIOM tJA" (6A" =N ·6{) .

.H,a.u.nM Terrepb o6lll,ee onpe,!l.e.1er-me ITOHHTHH o6 n,u:eaJihHbiX CBH3HX, KOTOpb!M Mbl Y}Ke ITOJib30BaJIHCb (eM. §

C6H.3U, 0/lH. JWmOpblX cyM.MQ B/leM.ehmapHblX pa6om UX peaKt{UU Ha AI060M B03MOOICHOAt nepeM.et.Uf!HUu cucme.Mbl paeHa HYAJO, T. e.

~6A~=0. (98)

flpuBe,ll.eHHoe B § 123 H BbipaJKeHHOe paseHCTBoM (52) ycJioBne H,!l.e- aJihHOCTH CB.H3eii, KOr)l.a OHH O)l.HOBpeMeHHO .HBJI.HIOTCH CTaiJ,HOH3pHbiMH, COOTBeTCTByeT onpe,!l.eJieHH!O (98), TaK KaK npH CTaiJ,HOHapHb!X

CBH3HX KaJK)l.Oe ,!l.eHCTBHTeJihHOe nepeMeru.eHHe COBIIa)l.aeT C O,!l.HHM H3 B03MOJKHhiX. ilo3TOMY npHMepaMH H.n_eaJibHb!X CBH3ell 6y,!l.yT BCe npH· Mephi, rrpHBe)l.eHHble B § 123.

.UJIH onpe.u.eJieHHH Heo6xo,!l.HMoro ycJioBHH paBHoBeCHH .u.oKaJKeM,

'ITO eCJIH

MeXaHHlJeCK3.H CHCTeMa

C H)l.eaJibHh!MH

CBH3HMH

H3XO)l.HTCH

non; ,ll.ellCTBHeM np~JIO.}KeHHbiX CHJI B paBHOBecHH,

TO npH

JII060M B03·

MOJKHOM

nepeMelll.eHHH CHCTeMhl

,ll.OJIJKHO BblllOJIHHTbCH paBeHCTBO

~6A~= 0

(99)

HJII1

~F~·brk = ~F~ 6sk cos ak = 0,

(99)

r)l.e ak - yroJI MeJK)l.y CHJIOH H B03MOJKHhiM nepeMelll.eHHeM.

360

0603H8'liiMpaBHO)l.eHCTBYIOill,IIe BCeX (!I BHeWHHX, II BHyTpeHHH"'S) aKTIIBHbiX CHJI II peaKIJ.liH CBH3eH, )l.eHCTBY!OliJ.IIX Ha KaKy!O-HII6y)l.h TO'l- Ky cHcTeMbi Bk, cooTBeTCTBeHHo qepe3 F't_ II Nk. Tor.na, nocKOJihKY Ka}l{)l.aH II3 TO'leKCHCTeMhr Haxo)l.IITCH B paaHoBeCHII, Fk +Nk=O, a c.rre-

.noaaTeJibHO, II CyMMa pa60T 3TIIX CII,TJ npii JI!060M nepeMelll,eHIIII TO'lKH

Bk 6y.neT To}l{e pasHa HYJI!O, T. e.

6A%+oAk =0. CocTaBIIB TaKHe pa-

BeHcTBa )l.JIH BCeX TO'leK CHCTeMbi

II CJIO}I{HB HX llO'lJieHHO, llOJIY'liiM

~<'>A~ +~<'>Ak = 0.

Ho TaK KaK CBH3H II.neaJibHhle, a <'>rk npe.ncTaBJIHIOT co6oi1: B03MO}I{Hhre nepeMelll,eHIIH TO'leKCHCTeMhi, TO BTOpaH CYMMa ITO yCJIOBIIIO (98) 6y,neT paBHa HYJIIO. Tor.na pasHa HYJIIO II nepBaH cyMMa, T. e. BhinOJIHHeTcH paBeHCTBO (99). TaKHM o6pa3oM, .noKa3aHo, 'lTOpaBeHcTBo (99) Bbrpa- }1{8eT HeOOXO)l.IIMOe yCJIOBHe paBHOBeCIIH CIICTeMbi.

DoKa}l{eM, 'lTO3TO ycJioarre HBJIHeTCH II .nocTaTO'lHhiM,T. e. 'lTOecJIII K TO'lK8MMeXaHII'leCKOH CHCTeMhl, HaXO!J.HIII.eHCH B llOKOe, npiiJIQ)!H!Tb aKTIIBHhie CHJibi tj,, y)l.OBJieTBOpH!OIIJ.He paBeHCTBy (99), TO CIICTeMa

OCTaHeTCH B TIOKOe. Ope)l.nO.TJO}I{II'\1o6paTHOe, T. e. 'lTOCHCTeMa npll 3T0\1

I1pii)l.eT B )l.BII}I{eHIIe !I HeKOTOpbie ee TO'lKHCOBeprnaT )l.eHCTBHTeJibHbie nepeMem.eHHH drn. Tor.na cnJihi PC coBeprnaT Ha 3THX nepeMeiiJ.eHIIHX pa60Ty H no TeopeMe 06 II3MeHeHHH KHHeTH'leCKOH3Hepnm 6y.neT:

dT = ~ dAk (dM = fi· drk),

r.ne, oqeBII)l,HO, dT>0, TaK KaK BHaqaJie CHCTeMa 6biJI8 B fiOKoe; CJie- )l.OBaTeJibHO, H ~dA~>O. Ho rrpii cTauHoHapHhiX CBH3HX .nefrcTBHTe.'lh-

Hbre nepeMeiiJ.eHHH drk COBll8,!1.8IOT C KaKHMH-TO H3 B03MO>I<Hb!X nepe:vleIIJ.eHHll 6rk II Ha 3TliX nepeMelll,eHIIHX TO}I{e )l.OJI/!{HO 6biTb ~OA~>O,

'lTOnpoTIIBopeqHT ycJIOBHIO (99). TaKHM o6pa30M, Kor.na npiiJIO}I{eHHhie CHJibi y)l.OBJieTBOpH!OT ycJIOBII!O (99), CHCTeMa II3 COCTOHHHH llOKOH BbiHTII He MO}I{eT II 3TO yCJIOBIIe HBJIHeTCH )l,OCT8TOqHhiM yc.TJOBHeM

paBHOBeCHH.

Ib .noKa3aHHoro BbiTeKaeT cJie.nyrom.nii: n p II H u II n B o 3 M o }1{-

H hi x n e p e M e m. e H II il *: OAJl paeHoeecuR .MexaHu<teawu cucmeMbt

c uoeaAbHbl.MU C6Jl3RMU He06XOOUM0 u oocmamO'iHO, 'im06bt CYMMa 3Ae-

.MeHmapHblX pa6om 8CeX OCUCmBYIOUI,UX Ha Hee aKmU6HbtX CUA ttpU A1060.M eos.MOJJCHO!rt nepe.MeU{eHuu cucme~rtbl 6btAa paeHa HYAIO. MaTeMaTII'leCKH

apopMyJIHpOB8HHOe yCJIOBIIe paBHOBeCIIH

Bbipa}l{aeTCH

paBeHCTBOM

(99), KOTOpoe H83biB8IOT T8K}I{e ypaBHemreM

B03M0}1{HblX

pa60T. 3TO

paBeHCTBO MO}I{HO em.e npe)l.CT8Bl1Tb B aHaJIIITH'leCKOHcpopMe (eM. § 87):

~ (F~x <'>xk + F~u 6yk + F~z ozk) = 0.

(100)

* B cpopMe, 6JH!3KOH K COBpeMeHHOH, HO 6e3 ,1\0Ka3aTeJibCTBa .9TOT npHHUKll :BDlCKa3aJI 3HaMeHHTDI'HMareMaTHK H M€XaHHK (IIIBeftuapeU 110 npOHCXOJK,1\€HHIO) l1oraHH BepuyJIJIH (1667-1748). B o6III.eM BHAe npHHUHTI BrrepBhie ccpopMyJiuposaJI H ,1\0KasaJI' )K. JlarpaHJK (1788 r.) 06o6JII.eHue npuHuuna Ha cJiyqafi Hey,nepJKuBaiOIII.HX c:sl!seji: DblJIO ,naHo M.B. OcTporpa.l\CKHM B pa6oTax 1838-1842 rr.

361

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